Find units of Z[w]

• January 27th 2008, 07:00 AM
Find units of Z[w]
Let $
w = \frac {-1}{2} + \frac {3^{1/2}}{2}i
$
, find units of Z[w].

1 is the unity, so it is a unit. And what else...?
• January 27th 2008, 08:11 AM
ThePerfectHacker
Quote:

Originally Posted by tttcomrader
Let $
w = \frac {-1}{2} + \frac {3^{1/2}}{2}i
$
, find units of Z[w].

1 is the unity, so it is a unit. And what else...?

The the previous excercises $a+bw$ is a unit if and only if $N(a+bw) = a^2+b^2 - ab = 1$. Thus, $a^2 - ab + (b^2 - 1) = 0$ we require that the discrimant, $b^2 - 4(b^2 - 4) > 0$, thus, $3b^2 < 4 \implies b=0,1,-1$, now you can solve for $a$.
• January 27th 2008, 09:23 AM
I don't understand how do you get to $
b^2 - 4(b^2 - 4) > 0
$
, is there a general rule for that or you did by algebra? Further, I worked this out and I have 3b^2 < 16, not 4, did I do something wrong?

Thanks.
• January 27th 2008, 12:59 PM
ThePerfectHacker
Quote:

Originally Posted by tttcomrader
I don't understand how do you get to $
b^2 - 4(b^2 - 4) > 0
$
, is there a general rule for that or you did by algebra? Further, I worked this out and I have 3b^2 < 16, not 4, did I do something wrong?

Thanks.

Yes. I made a mistake it should be 3b^2 < 16. It is just the discrimant of the quadradic.